basis-map coefficients of A scaled Hermite triangle.

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Triangle begins:
   1: 0, 0, 1;
   2: 0, 0, 0, 1;
   3: 0, 0, 0, 0, 1;
   4: 0, 0, 0, 0, 0, 1;
   5: 0, 0, 0, 0, 0, 0, 1;
   6: 0, 0, 0, 0, 0, 0, 0, 1;
   7: 0, 0, 0, 0, 0, 0, 0, 0, 1;
   8: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  10: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

The row polynomials satisfy P(n,x) = x*P(n-1,x) for n >= 2, with P(1,x) = x^2.

P[1, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = x^2; for(n = 2, N, my(i = n); v[i] = x*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-470643e70af29063
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-470643e70af29063, e-f9cce4eb6656235e
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-470643e70af29063, e-f9cce4eb6656235e
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f9cce4eb6656235e

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• definition_equality e-8874f9d1eed458eb root: theorem check

  Ore-algebra identity of two definitions polytool ?

• parent_provers periodic_product e-470643e70af29063 root: theorem check

  Parent-row provers of polytool polytool ?

• preserver_image mobius, shift 0 e-f9cce4eb6656235e

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)

Used in 15 proof steps

• e-33def10999a9e794 basis_map (A013610) → basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.
• e-66762082773dd3e0 basis_map (A008277) → basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.
• e-77b4dff64dd0f71c basis_map (A112227) → A scaled Hermite triangle.
• e-7bee75fc05394bb8 basis_map (A013609) → basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.
• e-b91bf4c52fd06745 basis_map (A013611) → basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.
• e-d82776c4e84d5954 basis_map (A007318) → shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.
• e-087ffb2de1c6b7b4 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of basis-map coefficients of A scaled Hermite triangle.
• e-35fb5901b32bd10b preserver_image (derivative_polynomial, shift 0) → derivative of basis-map coefficients of A scaled Hermite triangle.
• e-695286cd251b7e4d preserver_image (reversal, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-972f6eabe8338070 preserver_image (euler_polynomial, shift 0) → theta_plus_one of basis-map coefficients of A scaled Hermite triangle.
• e-c527a11d264443d6 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.
• e-cb9b2e4b7b317e85 preserver_image (affine, shift 0) → shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.
• e-f9cce4eb6656235e preserver_image (mobius, shift 0) → basis-map coefficients of A scaled Hermite triangle.
• e-56bb291f907074b7 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of basis-map coefficients of A scaled Hermite triangle.
• e-fc5f49d4b55941b1 veronese_section (r=2, j=1, shift 0) → veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomia…

Related sequences ?

Same fingerprint (75)

Equal up to content, sign, reversal and a small index shift.

• A178820 → γ
• basis-map coefficients of A026729, shifted → (θ+1)²
• A023531 → D → (θ+1)²
• basis-map coefficients of A112227 → D
• A136572 → θ+1
• A156992 → γ
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A136572 → D
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• A023531 → 2D+1+2n → Veronese 2
• A038325 → γ
• A178821 → γ
• A130123 → θ+1
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A344565 → γ
• A196347 → γ
• A010054 → reverse → θ+1
• basis-map coefficients of A021012 → D
• A010054 → reverse → reverse
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
• basis-map coefficients of A026729, shifted → D
• A010054 → reverse → reverse
• basis-map coefficients of A112227 → (θ+1)²
• A010054 → reverse → D
• basis-map coefficients of A297191, shifted → D
• A010054 → reverse → D
• A023531 → (θ+1)² → θ+1
• A010054 → reverse → θ+1
• A010054 → reverse → (θ+1)²
• A023531 → (θ+1)² → reverse
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
• basis-map coefficients of A030528, shifted → reverse
• A010054 → reverse → D
• basis-map coefficients of A297191, shifted → θ+1
• basis-map coefficients of A030528, shifted → θ+1
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• A028326 → γ
• A135838 → γ
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• basis-map coefficients of A026729, shifted → θ+1
• basis-map coefficients of A030528, shifted → D
• basis-map coefficients of A026729 → (θ+1)²
• A023531 → θ+1
• A023531 → (θ+1)² → (θ+1)²
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• A130123 → reverse
• basis-map coefficients of A112227 → θ+1
• basis-map coefficients of A021012 → θ+1
• basis-map coefficients of A030528, shifted → (θ+1)²
• A130123 → (θ+1)²
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A023531 → (θ+1)² → D
• A133080 → Veronese 2
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• A130123 → D
• A131431: 3n + 1 preceded by n zeros.
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• A178822 → γ
• A010054 → reverse → (θ+1)²
• basis-map coefficients of A021012 → (θ+1)²
• A085881 → γ
• A038338 → γ
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• basis-map coefficients of A297191, shifted → (θ+1)²
• A356546 → γ
• basis-map coefficients of A026729, shifted → reverse
• A023531 → 2D+1+2n → Veronese 2 (j=1)
• A010054 → reverse → θ+1
• A136572 → (θ+1)²
• A007318 → D → γ
• A038221 → γ
• basis-map coefficients of A026729 → D
• basis-map coefficients of A021012 → reverse